A099529 -id:A099529 - cp's OEIS frontend

A050935 a(n) = a(n-1) - a(n-3) with a(1)=0, a(2)=0, a(3)=1.

0, 0, 1, 1, 1, 0, -1, -2, -2, -1, 1, 3, 4, 3, 0, -4, -7, -7, -3, 4, 11, 14, 10, -1, -15, -25, -24, -9, 16, 40, 49, 33, -7, -56, -89, -82, -26, 63, 145, 171, 108, -37, -208, -316, -279, -71, 245, 524, 595, 350, -174, -769, -1119, -945, -176, 943, 1888, 2064, 1121, -767, -2831, -3952

Offset: 1

Richard J. Palmaccio (palmacr(AT)pinecrest.edu), Dec 31 1999

The Ze3 sums, see A180662, of triangle A108299 equal the terms of this sequence without the two leading zeros. - Johannes W. Meijer, Aug 14 2011

R. Palmaccio, "Average Temperatures Modeled with Complex Numbers", Mathematics and Informatics Quarterly, pp. 9-17 of Vol. 3, No. 1, March 1993.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
José L. Ramírez, Víctor F. Sirvent, A note on the k-Narayana sequence, Annales Mathematicae et Informaticae, 45 (2015) pp. 91-105.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1).

When run backwards this gives a signed version of A000931.
Cf. A099529.
Apart from signs, essentially the same as A078013.
Cf. A203400 (partial sums).

a050935 n = a050935_list !! (n-1)
a050935_list = 0 : 0 : 1 : zipWith (-) (drop 2 a050935_list) a050935_list
-- Reinhard Zumkeller, Jan 01 2012

A050935 := proc(n) option remember; if n <= 1 then 0 elif n = 2 then 1 else A050935(n-1)-A050935(n-3); fi; end: seq(A050935(n), n=0..61);

LinearRecurrence[{1,0,-1},{0,0,1},70] (* Harvey P. Dale, Jan 30 2014 *)

a(n)=([0,1,0; 0,0,1; -1,0,1]^(n-1)*[0;0;1])[1,1] \\ Charles R Greathouse IV, Feb 06 2017

From Paul Barry, Oct 20 2004: (Start)
G.f.: x^2/(1-x+x^3).
a(n+2) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k)*(-1)^k. (End)
G.f.: Q(0)*x^2/2, where Q(k) = 1 + 1/(1 - x*(12*k-1 + x^2)/( x*(12*k+5 + x^2 ) - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 12 2013

Offset adjusted by Reinhard Zumkeller, Jan 01 2012

A099531 Expansion of (1+x)^3/((1+x)^3+x^4).

Original entry on oeis.org

1, 0, 0, 0, -1, 3, -6, 10, -14, 15, -7, -20, 80, -188, 351, -549, 702, -622, -42, 1839, -5471, 11560, -20064, 29144, -33329, 21059, 27730, -142182, 355626, -689121, 1114937, -1490892, 1461360, -337220, -2996465, 10030587, -22226506, 39921442, -60118930, 72788383, -55703295, -31057776

Offset: 0

Paul Barry, Oct 20 2004

Binomial transform is A099530.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3,-3,-1,-1).

Cf. A099529.

CoefficientList[Series[(1+x)^3/((1+x)^3+x^4),{x,0,50}],x] (* or *) LinearRecurrence[{-3,-3,-1,-1},{1,0,0,0},50] (* Harvey P. Dale, Feb 21 2016 *)

a(n)=-3a(n-1)-3a(n-2)-a(n-3)-a(n-4); a(n)=sum{j=0..n, sum{k=0..floor(j/4), C(n, j)(-1)^(n-j)C(j-3k, k)(-1)^k}}.

A020713 Pisot sequences E(5,9), P(5,9).

Original entry on oeis.org

5, 9, 16, 28, 49, 86, 151, 265, 465, 816, 1432, 2513, 4410, 7739, 13581, 23833, 41824, 73396, 128801, 226030, 396655, 696081, 1221537, 2143648, 3761840, 6601569, 11584946, 20330163, 35676949, 62608681, 109870576, 192809420, 338356945, 593775046, 1042002567

Offset: 0

David W. Wilson

nonn

Colin Barker, Table of n, a(n) for n = 0..1000
Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1).

This is a subsequence of A005314.
See A008776 for definitions of Pisot sequences.
Cf. A099529.

Iv:=[5, 9]; [n le 2 select Iv[n] else Ceiling(Self(n-1)^2/Self(n-2)-1/2): n in [1..40]]; // Bruno Berselli, Feb 04 2016

RecurrenceTable[{a[0] == 5, a[1] == 9, a[n] == Ceiling[a[n - 1]^2/a[n - 2]-1/2]}, a, {n, 0, 40}] (* Bruno Berselli, Feb 04 2016 *)
LinearRecurrence[{2,-1,1},{5,9,16},40] (* Harvey P. Dale, Aug 03 2021 *)

lista(nn) = {print1(x = 5, ", ", y = 9, ", "); for (n=1, nn, z = ceil(y^2/x -1/2); print1(z, ", "); x = y; y = z;);} \\ Michel Marcus, Feb 04 2016

a(n) = 2*a(n-1) - a(n-2) + a(n-3) (holds at least up to n = 1000 but is not known to hold in general).
Empirical g.f.: (5-x+3*x^2) / (1-2*x+x^2-x^3). - Colin Barker, Jun 05 2016
Theorem: E(5,9) satisfies a(n) = 2 a(n - 1) -  a(n - 2) + a(n - 3) for n>=3. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - N. J. A. Sloane, Sep 09 2016
a(n) = (-1)^n * A099529(n+6). - Jinyuan Wang, Mar 10 2020

A334293 First quadrisection of Padovan sequence.

Original entry on oeis.org

1, 0, 2, 5, 16, 49, 151, 465, 1432, 4410, 13581, 41824, 128801, 396655, 1221537, 3761840, 11584946, 35676949, 109870576, 338356945, 1042002567, 3208946545, 9882257736, 30433357674, 93722435101, 288627200960, 888855064897, 2737314167775, 8429820731201, 25960439030624

Offset: 0

Oboifeng Dira, Apr 21 2020

			For n=3, a(3) = 2*a(2) + 3*a(1) + a(0) = 2*2 + 3*0 + 1 = 5.

Colin Barker, Table of n, a(n) for n = 0..1000
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (2,3,1).

Quadrisection of A000931.
Bisection (even part) of A099529.
Cf. A099098, A091140.

Vec((1 - 2*x - x^2) / (1 - 2*x - 3*x^2 - x^3) + O(x^30)) \\ Colin Barker, Apr 27 2020

a(n) = A000931(4n).
a(n) = A099529(2n).
a(n) = Sum_{k=0..n} binomial(2*n-k-1, 2*k-1).
a(n) = 2*a(n-1)+3*a(n-2)+a(n-3), a(0)=1, a(1)=0, a(2)=2 for n>=3.
G.f.: (1 - 2*x - x^2) / (1 - 2*x - 3*x^2 - x^3). - Colin Barker, Apr 27 2020

A052625 E.g.f. (1-x)^2/(1-2x+x^2-x^3).

Original entry on oeis.org

1, 0, 0, 6, 48, 360, 3600, 45360, 645120, 10160640, 177811200, 3432844800, 72329241600, 1650160512000, 40537905408000, 1067062284288000, 29961435119616000, 893842506805248000, 28234468042260480000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 571

spec := [S,{S=Sequence(Prod(Z,Z,Z,Sequence(Z),Sequence(Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[(1-x)^2/(1-2x+x^2-x^3),{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, May 22 2012 *)

E.g.f.: -(-1+x)^2/(-1+2*x-x^2+x^3)
Recurrence: {a(1)=0, a(0)=1, a(2)=0, (-11*n-6-n^3-6*n^2)*a(n) +(n^2+5*n+6)*a(n+1) +(-2*n-6)*a(n+2) +a(n+3)=0}
Sum(-1/23*(2-11*_alpha+6*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(-1+2*_Z-_Z^2+_Z^3))*n!
a(n) = (-1)^n*n!*A099529(n). - R. J. Mathar, Jun 03 2022

cp's OEIS Frontend

A050935 a(n) = a(n-1) - a(n-3) with a(1)=0, a(2)=0, a(3)=1.

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A099531 Expansion of (1+x)^3/((1+x)^3+x^4).

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A020713 Pisot sequences E(5,9), P(5,9).

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A334293 First quadrisection of Padovan sequence.

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A052625 E.g.f. (1-x)^2/(1-2x+x^2-x^3).

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